THE JOURNAL OF CHEMICAL PHYSICS 134, 094103 (2011)

Transition-metal dioxides: A case for the intersite termin Hubbard-model functionals

Heather J. Kulik1,a) and Nicola Marzari21Department of Chemistry, Stanford University, Stanford, California 94305, USA2Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom

(Received 16 November 2010; accepted 7 February 2011; published online 2 March 2011)

Triatomic transition-metal oxides in the “inserted dioxide” (O–M–O) structure represent one of thesimplest examples of systems that undergo qualitative geometrical changes via subtle electronic-structure modulation. We consider here three transition-metal dioxide molecules (MO2 where M= Mn, Fe, or Co), for which the equilibrium structural (e.g., bent or linear geometry) and electronic(e.g., spin or symmetry) properties have been challenging to assign both theoretically and experi-mentally. Augmenting a standard density-functional theory (DFT) approach with a Hubbard term(DFT+U ) occasionally overlocalizes the 3d manifold, leading to an incorrect bond elongation and,in turn, poor equilibrium geometries for MO2 molecules, while preserving good spin-state split-tings. Proper description of both geometry and energetics for these molecules is recovered; however,through either calculating DFT+U relaxations at fixed M–O bond lengths or by inclusion of an in-tersite interaction term V that favors M(3d)–O(2p) interactions. In this latter case, both U and V arecalculated fully from first-principles and are not fitting parameters. Finally, we identify an approachthat more accurately determines the Hubbard U over a coordinate in which the covalent character ofbonding varies. © 2011 American Institute of Physics. [doi:10.1063/1.3559452]

I. INTRODUCTION

Transition-metal oxides are fundamental reactive inter-mediates in both biological systems like metalloenzymes1–3

and in inorganic catalysts.4–6 The triatomic transition-metaloxides that consist of a single transition metal and two oxygenatoms represent a significant increase in complexity over thesimplest case of the diatomic molecule. In particular, there areseveral MO2 isomers: one in which the dioxygen molecule in-teracts weakly with the transition-metal in a side-on or end-ongeometry or one in which the metal forms an inserted diox-ide in either a bent or linear fashion.7 We focus here on thelatter case, which has been shown to be more energeticallystable for most transition-metal-containing molecules.8 Theinserted dioxides are of particular interest because the relativeenergetics of a bent or linear structure may be very sensitiveto the spin, symmetry, and nature of hybridization in a givenelectronic state.

A large number of experimental studies of inserteddioxides have been carried out on the mid-row transition-metal dioxides, MnO2,9–11 FeO2,12–14 and CoO2.15–18 Ex-perimentally, the early- to mid-row transition-metal dioxidesprefer bent structures, while the later transition-metal ox-ides have linear structures. The mid-row transition-metal ox-ides represent a formidable challenge for theoretical calcula-tions, which, while extensive for MnO2,19, 20 FeO2,8, 20–22 andCoO2,20, 23, 24 have been inconclusive or contradictory with re-spect to experiment. The greatest difficulty for this class ofsystems from a computational standpoint is to correctly assignthe ground state spin and symmetry as well as the geometricstructure (i.e., whether the molecule is bent or linear).

a)Electronic mail: hkulik@stanford.edu.

We have previously shown that augmenting a standarddensity-functional theory (DFT) approach with a Hubbard-like “+U” term can greatly improve spin-state assignmentsin transition-metal-containing molecules. However, the sameDFT+U method does not always improve structural propertiesover standard DFT (we use here generalized-gradient approx-imations, referred to as GGAs).25 In this work, we exploreseveral means to correctly describe both structure and ener-getics for this class of molecules.

The DFT+U method includes an approximation basedupon the Hubbard model,26, 27 and it has been widely usedin the solid state physics community to treat strongly corre-lated systems.28–30 We have recently demonstrated4, 5, 25 thatsuch an approach can very accurately treat transition-metalcomplexes,31, 32 where only one or few transition-metals areinvolved (see Ref. 33 for a comparison to different func-tionals). DFT+U achieves high accuracy by correcting self-interaction errors of standard local or semilocal exchange-correlation functionals with Hartree–Fock-like treatments ona localized set of atomic orbitals. In addition, the strengthof this local correction can be determined fully from first-principles, thanks to a linear-response formulation.34 A rota-tionally invariant formulation of DFT+U (Refs. 4,34, and 35)adds a Hubbard term of the form

EU = U

2

∑I,σ

Tr[nIσ (1 − nIσ )], (1)

where nIσ is the occupation matrix of the localized mani-fold(s) at site I with spin σ . This functional form, whichis tied to the exact correction needed for simple exchange-correlation functionals in the limit of an atomic system,34, 36

penalizes fractional occupations and approaches zero as

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094103-2 H. J. Kulik and N. Marzari J. Chem. Phys. 134, 094103 (2011)

n approaches 0 or 1. Recently introduced and relateddensity-functional theory approaches aim to improve delo-calization errors through fixing the energetics at fractionaloccupations,37 increasing the order of approximation in theexchange-correlation functional,38 localizing orbitals in anad hoc fashion,39 and varying the amount of Hartree–Fockexchange in the functional based on a distance constraint.40, 41

Occupations of localized atomic levels in a molecule thatenter into EU may typically be determined through projec-tions onto an atomic or molecular orbital basis, populationanalysis, or any other number of methods.34 We project ontoatomic orbitals because of their simplicity, generality and be-cause they permit comparison across many different coordi-nation environments and structures. Generally, all approachesfor determining occupations lead to the same expression forthe occupation matrix elements:

nIσmm ′ =

∑k,v

f σkv〈�σ

kv | P Imm ′ | �σ

kv〉, (2)

where �σkv is the valence electronic wavefunction correspond-

ing to a given molecular state with spin σ and f σkv is the

occupation of the molecular level. The P Imm ′ are generalized

projection operators that satisfy a number of conditions34 in-cluding that the orthogonality of projectors for a single site ismaintained.35

In molecules, the Hubbard augmentation of the func-tional improves the delocalization that occurs as a result ofthe self-interaction error present in most functionals. It hasbeen shown34 that U may be calculated directly from the lin-ear response:

χI J = ∂2 E

∂αI αJ= ∂nI

∂αJ, (3)

where χI J is the response function obtained from applying anarbitrary shift αJ to the potential on the site J that results ina reorganization of the occupations nI on site I . A similar ex-pression may be obtained for the noninteracting case, χ0, asa linear shift in the potential can still result in rehybridizationthat must be removed from our overall expression to deter-mine U . Our final value is then obtained as

U = χ−10 − χ−1, (4)

χ is simply a scalar if we are only interested in a single man-ifold and site or it becomes a matrix in the case of multiplesites or manifolds. We stress that U is calculated fully fromfirst-principles as a system-dependent property and not usedas a fitting parameter in any way.

A recent extension to this approach focuses on inclusionof intersite interactions through an additional “+V” term.42

The intersite interactions play an important role in condensedphase systems where both extended and localized 3d-derivedstates coexist. In DFT+U+V, interactions between electronson different sites (e.g., a transition metal’s 3d states and anoxygen atom’s 2p states) are included in the model Hamil-tonian. It follows that intersite interactions are particularlyrelevant in highly covalent transition-metal complexes. The

simplified additive form of the “+U +V” energy functional is

EU V =∑I,σ

U I

2Tr[nI Iσ (1 − nI Iσ )] −

∗∑I,J,σ

V I J

2Tr[nI Jσ nJ Iσ ],

(5)where the first term is simply the same as the original EU

functional and the second term is the additional intersite con-tribution. The “+V” term sums the intersite interactions overeach atom I and any atom J within a given distance (indi-cated by the ∗ index). For a relatively small molecule, anyatom of the same type is typically counted in the intersite in-teraction, while for larger molecules a distance cutoff may beimposed. While in the original “+U” functional, fractional oc-cupations were penalized, the new “+V” term stabilizes statesthat are formed from combinations of atomic orbitals on dif-fering atoms. The result is a competition between “+U” and“+V” terms that effectively favors the covalent nature of bondsthat may be overly penalized by a straightforward DFT+U ap-proach. The V term that enters into the functional may alsobe calculated alongside U in a similar manner. A more de-tailed description of the intersite extension and calculation ofthe V term is provided in Ref. 42. For molecules, we willshow that the “+V” term primarily reverses some of the bondelongation that occurs in DFT+U, thus improving structuraldescriptions, while maintaining the energetic state splittingsthat make DFT+U often outperform a standard DFT(GGA)approach.

II. METHODS

Plane-wave, density-functional calculations were com-pleted with the QUANTUM-ESPRESSO package43 using thePerdew–Burke–Ernzerhof44 GGA. We augmented this stan-dard GGA functional with a self-consistent, linear-response,Hubbard U term, as previously outlined4, 34 as well as with arecently introduced “+V” term.42 Ultrasoft pseudopotentialswere used with a plane-wave cutoff of 30 Ry for the wave-function and 300 Ry for the charge density to ensure forcesand spin-state splittings were converged with respect to ba-sis set size. In the case of Mn and Fe, the semicore 3s and3p states were included in the valence, while for Co only 4sand 3d states were treated as valence states. DFT+U calcula-tions were carried out with a U on 3d states of the transition-metal, while the intersite calculations included both a U on3d states and an inter-site term, V between the 3d states ofthe metal and the 2p states of oxygen. The V term was cal-culated by perturbing the oxygen 2p states and calculatingthe linear response of the 3d manifold from a matrix of boththe 2p and 3d response functions. Damped dynamics with ageometric constraint were used to obtain single-dimensionalpotential energy curves for the angle-dependent energetics ofthe transition-metal dioxides. These calculations optimizedthe metal-oxo bond lengths with respect to a fixed O–M–Oangle that was resolved in 5◦ increments.

III. RESULTS

The transition-metal center in the inserted dioxide formof MO2 (M = Mn, Fe, Co) may be approximately described

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094103-3 DFT+U and DFT+UV on transition-metal dioxides J. Chem. Phys. 134, 094103 (2011)

TABLE I. Calculated Hubbard U from linear-response (U0) and self-consistent calculation (Uscf) for relaxed MnO2, FeO2, and CoO2 structuresin the lowest-lying spin states (M refers to multiplicity). The spin state cor-responding to the experimental ground state is indicated by a bold typeface.The last line in the table is an average over all spin states.

MnO2 FeO2 CoO2

Bent Linear Bent Linear Bent LinearM U0 Uscf U0 Uscf M U0 Uscf U0 Uscf M U0 Uscf U0 Uscf

2 4.5 4.6 7.0 7.2 1 5.0 5.5 7.2 7.0 2 6.4 6.5 6.6 6.54 6.0 5.9 7.0 7.1 3 6.0 6.6 7.0 7.5 4 4.5 4.5 5.5 5.76 4.7 4.7 6.0 6.1 5 4.6 5.1 5.4 6.1 6 4.6 4.6 5.3 5.3

5.1 5.1 6.7 6.8 5.2 5.7 6.5 6.8 5.2 5.2 5.8 5.8

as being in a high-valent M(IV) state. However, the moleculesexhibit a strongly covalent nature, and a O2−-M4+-O2− pic-ture is overly simplistic. In fact, for manganese, iron, andcobalt, a variety of MO2 states exist within 1 eV of the lowestenergy structure, in contrast to the larger 2–4 eV splittings ofatomic levels of M4+ ions.45 We calculate the linear-responseand self-consistent values of Hubbard U for both bent andlinear structures of each of the lowest energy spin states inTable I.

Manganese, iron, and cobalt all yield similar values ofU for the bent structures at around 5 eV. The linear structureexhibits a slightly larger value of U , but we will later showthat the 6–7 eV value is, in part, an overestimate. The calcu-lation of the intersite V was also carried out for each speciesand found to be, on average, 2 eV. The small spread in val-ues of Hubbard U for each spin state and structure suggeststhat by employing an average U and V in all GGA+U andGGA+U+V calculations that all spin states and moleculeswill be treated equivalently well. The experimental groundstate spin of each molecule is highlighted, and this spin stategenerally corresponds to a lower multiplicity than that of theisolated M(IV) ion.45 We will later consider which orbitalscontribute to favoring bent structures over linear structures ineach of these molecules.

A. MnO2

Numerous experimental9–11 and theoretical19, 20 studieshave been carried out on manganese dioxide. Early electronspin resonance experiments suggested that the inserted diox-ide was a linear molecule,9 but later experimental infraredspectra that were collected using laser-ablated manganeseatom precursors instead demonstrated more conclusively thatthe OMnO structure was a bent 4 B1 state with a 135 ± 5◦

O–Mn–O angle.10 Bond angles of inserted dioxide moleculescan be determined experimentally from infrared (IR) spectra.While the symmetric stretching mode (v1) is IR inactive forthe linear structure, use of isotopes as in 18OM16O will reducethe symmetry and make the symmetric-stretch IR active. Bycomparing the spectra with and without isotopic substitution,one can determine if the molecule is linear. The isotopic ra-tio of the antisymmetric stretching mode frequency providesa route to estimate the bond angle.7 An upper limit is obtainedfrom the oxygen isotopic ratio and a lower limit is obtained

TABLE II. Comparison of experimental bond angles for MO2 (M = Mn(Ref. 10, 19), Fe (Ref. 13, 14), Co (Ref. 17, 18)) to calculated DFT(GGA),DFT+U , DFT+U |r0 (calculated at the GGA M–O internuclear separation),and DFT+U+V values.

State DFT +U +U |r0 +U+V Expt.4 B1 MnO2 128 180 140 143 135 ± 5

3 B1 FeO2 138 180 155 156 150 ± 102�+

g CoO2 158 180 180 180 180

from the metal isotopic ratio. The bond angle determinedfrom the average of the two limits of the asymmetric stretchfrequency has been shown to be very close to the true bondangle for simple cases including SO2 (Ref. 46) and S3.47, 48

A comparison of the experimental and theoretical bond an-gles for all molecules considered in this work are presented inTable II. It is worth noting that a molecule may appear quasi-linear if the linear structure is only slightly higher in energythan the bent minimum energy structure because the experi-mental bond angle is determined from a vibrational averageat the zero point level.

Our theoretical calculations show that, along the GGApotential energy surface, a bent 4 B1 MnO2 inserted dioxide isstabilized by about 0.45 eV with respect to the linear structuresaddle point. The 4 B1 state corresponds to an electronic con-figuration of 11a1

110a111a2

24b116b2

2, in agreement with previoustheoretical results.19 For GGA+U calculations with increas-ing values of Hubbard U, the relative energetics of the bentand linear structures are swapped, and the linear structure isinstead favored at U = 6 eV. The bent and linear structuresbecome degenerate in energy at U = 4.6 eV, and the splittingat larger values of U is quite small—only around 0.1 eV. Em-ploying a GGA+U approach has the overall effect of flatten-ing the angular-dependent potential energy curve for MnO2

from a double-welled structure to a single, wider well with alinear structure at the minimum, which does not agree withthe experimentally determined structure.

Upon observing that the GGA+U preference for linearstructures also corresponded to a Mn–O bond distance elon-gated by nearly 0.1 Å with respect to GGA (Table II), weinvestigated how angular potential energy curves might varyat differing fixed Mn–O bond distances. A comparison of theangular dependence of the energetics for different fixed val-ues of rMn−O reveals that a bent global minimum is preservedeven for values of U as high as 6 eV if rMn−O is fixed to 1.55 Å(Fig. 1), which is approximately the GGA equilibrium value(see Table III). At a slightly larger rMn−O = 1.70 Å, both GGAand U = 4 eV curves exhibit a bent minimum energy struc-ture. For the stretched rMn−O = 1.85 Å, all values of U find

TABLE III. Comparison of average bond lengths (M–O in Å) for MO2 (M= Mn, Fe, Co) corresponding to minimum energy configurations along theθ -coordinate from DFT(GGA), DFT+U , and DFT+U+V .

State DFT +U +U+V4 B1 MnO2 1.61 1.70 1.59

3 B1 FeO2 1.59 1.67 1.582�+

g CoO2 1.55 1.63 1.56

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094103-4 H. J. Kulik and N. Marzari J. Chem. Phys. 134, 094103 (2011)

110 130 150 1700.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8R

elat

ive

ener

gy (

eV)

U = 0U = 4U = 6

110 130 150 170 110 130 150 170

O-Mn-O Angle (°)O-Mn-O Angle (°)O-Mn-O Angle (°)

rMn-O = 1.55Å rMn-O = 1.70Å rMn-O = 1.85Å

FIG. 1. Comparison of angular dependence of the energy for constant valuesof rMn−O = 1.55, 1.70, and 1.85 Å (left to right graphs) for different valuesof Hubbard U . The angular-dependent potential energy curves depicted arefor GGA (U = 0 eV, black circles) and GGA+U (U = 4 eV, blue trianglesand U = 6 eV, red squares), where the two GGA+U values correspond to alower value of U at which the bent structure is preferred over linear structureat equilibrium bond length and a higher value of U where the trend reverses.

a linear structure to be lowest in energy. This trend demon-strates that the preference of a GGA+U approach for linearstructures over bent structures arises because of the naturaltendency to overelongate bonds, particularly when the occu-pation matrix is obtained from projections onto atomic or-bitals (as opposed to bond-centered orbitals). This bond elon-gation has a secondary effect of further weakening the an-gular energetic dependence in addition to the weaker depen-dence for GGA+U curves over GGA at a fixed r . Constrain-ing the bond distance in GGA+U calculations to GGA values(referred to as GGA+U |r0 ) recovers a bent structure with anO–Mn–O angle of 140◦ in excellent agreement with the 135± 5◦ experimental value.10

As an alternative to constraining the GGA+U cal-culation to GGA bond distances in order to determinethe O–Mn–O angle, we calculated properties of 4 B1

MnO2 with GGA+U+V . Recently implemented GGA+U+Vapproaches42 include an intersite terms for interactions be-tween the 3d states of Mn with the 2p states of O. Inclusion ofa V = 2 eV in addition to the Uscf = 5 eV recovers the correctbent geometry of quartet MnO2. The Mn–O bond distance de-termined from a full relaxation with GGA+U+V is essentiallythe same as that for GGA (see Table III). Additionally, the O–Mn–O angle is 139◦ in agreement with experiment and thepreviously determined GGA+U |r0 value. Both approaches ef-fectively accomplish the same correction over standard GGA:they reduce Mn–O bond distances and, in doing so, correctlyfavor the experimentally observed bent geometry.

The lowest-lying state of MnO2, 4 B1, is well-separatedfrom other electronic states for all methods considered. Thenext lowest states, 2 B1 and 6 A1, lie 0.7 and 2.0 eV above theground state, respectively, when calculated with GGA. Theseenergetic separations agree with previous GGA results,19 andthey do not shift qualitatively with GGA+U or GGA+U+V .The 2 B1 state differs from 4 B1 by a flipped spin for anelectron in an a1 molecular orbital. As a result, the doubletstate geometric structure is comparable to that of the quar-

TABLE IV. Comparison of some bond angles (in degrees) and bond lengths(in Å) for excited states of MO2 (M = Mn or Fe) calculated with DFT(GGA),DFT+U , and DFT+U+V .

State DFT +U +U+V2 B1 MnO2 1.62,131 1.66,154 1.61,1376 A1 MnO2 1.68,105 1.75,120 1.69,111

5 B2 FeO2 1.61,118 1.64,131 1.60,1201 A1 FeO2 1.57,141 1.61,180 1.58,148

tet, with GGA bond length and angle equal to 1.62 Å and131◦, respectively. The GGA+U (U = 5 eV) structure is stillbent, though slightly less, contrary to the results obtainedfor the quartet. The GGA+U |r0 and GGA+U+V structuralproperties are intermediate between GGA+U and GGA (seeTable IV). The 6 A1 state has a much more strongly bent min-imum energy structure, with a GGA bond angle around 105◦.This structure becomes flatter, though still linear, in GGA+Uwith a bond angle of 120◦ (see Table IV). We will later showthat the sharper bond angle of the sextet state is directly de-rived from enhanced occupation of Mn-derived δ orbitals.

B. FeO2

Numerous theoretical8, 20–22 and experimental12–14 stud-ies have been carried out on iron dioxide. Formation of theOFe(IV)O inserted dioxide is believed to occur experimen-tally via formation of an excited peroxo Fe(O2) species whichisomerizes to form the more stable inserted dioxide.8 Theoret-ical calculations22 on the inserted dioxide species of FeO2 inseveral spin states have demonstrated that both pure and hy-brid GGAs perform reasonably well at assigning relative en-ergetics and structures of the lowest-lying triplet and quintetstates. Unlike the manganese species, the iron dioxide tripletand quintet states are very close in energy, and determinationof the relative energetics of these states can be very sensitiveto the approach employed.

Experimentally, the 3 B1 state is believed to be the lowestin energy, and its bond angle has been measured to be 150± 10◦. The ground state determined by GGA calculations isthe quintet, not the triplet, though the triplet is only 0.07 eVhigher in energy. The triplet structure in GGA is just below theexperimental angle uncertainty at about 138◦. The 5 B2 stateof FeO2 has a sharper O–Fe–O bond angle than the tripletby at least 20◦ for all methods employed. The experimentallydetermined bond angle agrees well with the calculated valuefor the triplet but not the quintet, making it unlikely that the

TABLE V. Bond distances (rCo–O, in Å), bond angles (∠ O–Co–O, in de-grees), and relative energies (Te in eV) for several states of CoO2. Columnslabeled as +V refer to including +V alongside +U .

rCo–O, ∠(O–Co–O) Te

State DFT +U +V DFT +U +V2�+

g (2 A1) 1.55,158 1.63,180 1.56,180 0.00 0.00 0.004 A1 1.58,123 1.61,148 1.57,128 0.07 0.12 0.126 A1 1.66,130 1.73,180 1.65,131 0.71 0.48 0.67

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094103-5 DFT+U and DFT+UV on transition-metal dioxides J. Chem. Phys. 134, 094103 (2011)

110 130 150 170

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0R

elat

ive

Ene

rgy

(eV

)

3B1

5B2

110 130 150 170 110 130 150 170

U = 0 V = 0 U = 5 V = 0 U = 5 V = 2

O-Fe-O Angle (°)O-Fe-O Angle (°)O-Fe-O Angle (°)

FIG. 2. Comparison of angular dependence of the energy for the two lowest-lying states of FeO2, 3B1 (black circles), and 5B2 (red squares) calculatedwith GGA (left), GGA+U (middle), and GGA+U+V (right).

quintet is the ground state as is predicted in GGA. A singletstate of FeO2 also lies around 0.5 eV above the triplet andquintet states but is well-separated from the triplet and quintetstates in the region of interest.

With GGA+U , we reverse the triplet–quintet splittingfrom GGA in improved agreement with experimental find-ings, but GGA+U also predicts a linear triplet (Fig. 2). Thegeometric transition from bent to linear with GGA+U isinduced, as in MnO2, by the Fe–O bond elongation from1.59 Å in GGA to 1.67 Å in GGA+U . Both the GGA+U |r0

and GGA+U+V approaches recover an O–Fe–O angle ofaround 155◦ in agreement with experiment. Both techniquesalso favor the triplet over the quintet by about 0.1–0.2 eV, inagreement with high-level, quantum chemistry approaches.22

Interestingly, the 5 B2 state remains bent for all methodsconsidered (see Table IV). Even though the energy differencebetween bent and linear structures is reduced from GGA (1.2eV) to only about 0.2 eV for GGA+U (and higher at about0.8 eV for GGA+U+V ), the difference in computed bondangle between all three approaches is only 15◦. Overall, aGGA+U |r0 or GGA+U+V result provides the best agreementfor both structure and energetics with experiment over GGAor GGA+U alone.

C. CoO2

Numerous experimental15–18 and theoretical20, 23, 24 cal-culations have been carried out on cobalt dioxide. Experi-mental studies including electron spin resonance,15 gas phaseflow reactivity,16 and infrared study of laser-ablated cobaltand oxygen in solid argon13, 18 have all pointed towards a2�+ ground state for the OCoO inserted dioxide. Density-functional theory calculations using the generalized-gradientapproximation, however, have failed to agree with experimen-tal observations by instead predicting a bent ground state onthe doublet spin surface.20, 23, 24 In fact, in some cases it wasobserved that higher spin states such as a sextet were stabi-lized in energy with respect to the experimentally observeddoublet state.24 However, combined coupled cluster theory

and experimental work later confirmed that a doublet statewith a linear structure is in fact preferred.18

Our GGA results find a doublet state to be the lowest by0.05 eV with respect to a quartet state and 0.7 eV with re-spect to the lowest sextet state. However, GGA predicts a bentstructure with an O–Co–O angle of 158◦ instead of a 2�+ lin-ear ground state. The GGA+U , GGA+U |r0 , and GGA+U+Vapproaches yield instead a linear structure in agreement withexperiment (see Table II). In contrast, the GGA excited 4 A1

state is bent with a 123◦ O–Co–O angle, which is also foundto be bent with the other approaches (Table V). GGA over-hybridizes the δ-like orbitals of Co and, as a result, favorsbent structures exclusively over linear structures. In the caseof doublet CoO2, both GGA+U and GGA+U+V descriptionprovide good agreement with experiment.

D. Overall view of hybridization in inserted dioxides

Although typically considered nonbonding, dxy anddx2−y2 atomic orbitals play a critical role in stabilizing a bentgeometry in inserted dioxides. A lobe of the transition metalorbital hybridizes with oxygen 2p orbitals to form a bondwith the majority of the density centered on the O–O midpoint(Fig. 3). If, however, we elongate the M–O bond, the distancebetween O 2p orbitals and dxy or dx2−y2 orbitals increases andoverlap decreases. For bond distances that are stretched (1.8and 2.0 Å) with respect to an equilibrium value of 1.6 Å, thishybridization that stabilizes a bent structure disappears. Sucha result explains the shallow angular dependence for elon-gated M–O bonds either through stretching or through use ofa “+U” approach.

The extent of deviation of the equilibrium bond angleof MO2 from 180◦ is not dictated by relative bond lengthsfor the different transition metal species. For all three transi-tion metals, the GGA bond lengths of the lowest energy states

FIG. 3. Depiction of molecular orbitals derived from Mn 3dx2−y2 and 3dxy

hybridization with O 2p states for three M–O bond distances (1.6, 1.8, and2.0 Å).

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094103-6 H. J. Kulik and N. Marzari J. Chem. Phys. 134, 094103 (2011)

are within about 0.06 Å of each other, and this spread is pre-served or reduced in GGA+U and GGA+U+V , respectively(Table III). Rather, distinctions between preferred O–M–Obond angles (Table II) are largely dictated by the relative pop-ulation of bonding and antibonding orbitals that enhance thedensity across the oxygen–oxygen distance as mediated pri-marily by δ-like orbitals on the metal (Fig. 3). The MnO2

species has the smallest O–M–O bond angle, while both FeO2

and CoO2 are significantly more obtuse at the GGA level,possibly because a lower electron count in Mn(IV) results inthe population of fewest antibonding orbitals. A side-by-sidecomparison of prototypical bonding orbitals between bent andlinear structures reveals that the 3dxy and 3dx2−y2 atomic or-bitals of the transition metal hybridize directly with oxygen2p orbitals to form bonds with density centered over the mid-point of the O–O distance when the molecule is bent. For lin-ear structures, the 3dxy and 3dx2−y2 orbitals instead are local-ized on the metal and do not participate in bonding. For latetransition metals, (i.e., Co and higher), one would predict fur-ther filling of the minority spin 3d levels populates nonbond-ing and antibonding orbitals that do not favor a bent structure.In fact, the early transition metals (Sc–Cr) have been shownexperimentally to have bent ground states, while the later tran-sition metals (Co–Zn) have all been shown to be linear.7

IV. CONCLUSIONS

We have studied the geometric and electronic structureof the inserted dioxide form of MO2 molecules (M = Mn,Fe, Co) using an extended Hubbard functional with both on-site U and intersite V interactions. Both U and V are deter-mined from first-principles, linear-response calculations thatmaintain the parameter-free nature of this approach. The ten-dency of a “+U”-only approach to overelongate bonds, cou-pled with the shallower angular features of MO2 moleculesas M–O is elongated, promotes a dramatic but predictablechange in structure that fails to agree with experiment forthese studies. While improvements upon GGA+U may be ob-tained simply through constraining M–O bond distances toGGA or experimental values, we showed that a generalization

that includes on-site and intersite interactions recovers withremarkable accuracy both structural and electronic proper-ties for the systems studied. This suggests that a generalized+U+V functional could be used successfully in cases withstrong covalent interactions.

We also showed that a counterintuitive increase in linear-response U for increasing O–Mn–O angles is explained asa consequence of reducing hybridization between Mn and Oand emptying of δ-like states in the minority spin manifold.By adjusting for this variation in occupancy, we obtained anadjusted linear-response U that is constant rather than increas-ing with increase in the O–Mn–O angle. In conclusion, thetriatomic MO2 transition-metal oxides represent an intrigu-ing test case for DFT-based electronic structure approaches;discrepancies resulting from adding a “+U” term have beenresolved and a new method for verifying the values obtainedfrom linear-response has been proposed.

APPENDIX A: REVISITING THE LINEAR-RESPONSEHUBBARD U

Calculated values of Hubbard U for linear structures areat least 1 eV higher than the Uscf values in the GGA bent ge-ometry for all three MO2 structures (see Table I). This trendis roughly independent of spin state and transition-metal iden-tity, but it is also counterintuitive. That is, we know that thelinear structure becomes more energetically stable for highervalues of U , and this lower energy corresponds to a reducedamount of hybridization and overall reduced sensitivity to anDFT+U approach with respect to the bent geometry. For theMnO2 case, increasing the O–Mn–O angle increases occupa-tions for the majority spin, while it simultaneously decreasesoccupations in the minority spin (Fig. 4). The decline in Mn3d minority spin density with increase in angle is also associ-ated with a corresponding increase in oxygen 2p and 2s den-sity. The response function declines more substantially withincreased O–Mn–O angle in the minority spin manifold thanfor the majority spin (Fig. 4). The primary source of reducedhybridization for MnO2 is in the minority spin manifold andis owed to the decreased occupation of δ-like orbitals, dxy anddx2−y2 , as the molecule becomes linear (Fig. 4).

0.5

0.6

0.7

0.8

0.9

1.0

Pro

ject

ed O

ccup

atio

ns (

n)

dxz/dyz

dxy/dx2−y

2

dz2

0.0

0.1

0.2

0.3

0.4

0.5n↑ n↓

110 130 150 170 110 130 150 170

O-Mn-O Angle (°) O-Mn-O Angle (°)

110 130 150 170110 130 150 170

−0.10

−0.08

−0.06

−0.04

−0.02

0.00

χ (e

− /

eV)

χ↑ χ↓

O-Mn-O Angle (°) O-Mn-O Angle (°)

dxz/dyz

dxy/dx2−y

2

dz2

FIG. 4. Comparison of spin up and spin down occupations (top) as well as response functions (bottom) for 3d manifold as O–Mn–O angle is varied. Theconverged (black lines) and bare (red lines) response functions are depicted on the same graphs. Average total responses and occupations are depicted as dashedlines.

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094103-7 DFT+U and DFT+UV on transition-metal dioxides J. Chem. Phys. 134, 094103 (2011)

110 130 150 170

5

6

7

8

9H

ubba

rd U

(eV

)UU↓U adj.U↓ adj.

O-Mn-O Angle (°)

FIG. 5. Dependence of linear-response U on O–Mn–O angle (in degrees) ascalculated by several means: the standard linear-response approach (magentasquares), only from the more partially occupied spin down manifold (U ↓,brown circles), and through a renormalization scheme that takes into accountthe emptying of spin down dxy and dx2−y2 orbitals as the O–Mn–O angleapproaches 180◦ (U , blue diamonds and U ↓, green triangles).

As a demonstration of the role that the declining occu-pation of minority spin δ-like orbitals play in the apparenttrend of increasing U with increasing O–Mn–O angle, we plotseveral representations of U in Fig. 5. By considering onlyχ ↓, we observe that the resulting value of U ↓ is more sen-sitive to angular variations than the total U , indicating thatthe minority spin plays a dominant role in the coordinate-dependence of linear-response U . We also plot in Fig. 5 anadjusted value of U that is defined by a coordinate-dependentresponse function, χ (r ):

χ (r ) =(∑

i

fi |max

fi (r )

) (∑i

χi

), (A1)

where r refers to any variable over which a reaction or coordi-nate is defined. The χi refer to individual response functionsfor each state i in the relevant manifold. A single referencepoint where all fi are maximal may often be chosen: in thiscase, an O–Mn–O angle of 100◦, is a point, r0, where boththe number of states and their fractional occupation are max-imally occupied along the reaction coordinate. A simplifiedapproximation for the adjusted U (Uadj) may be made if weonly consider occupation variations for the states that are oc-cupied at some point along the coordinate but unoccupied atother points:

Uadj ≈ UNocc,min + fvar(r )

fvar,max

Nocc,max, (A2)

where Nocc refers to the number of states that are occupiedeither maximally or minimally (in the case of total linear-response U , these numbers are 10 and 8 respectively). Theratio fvar(r )/ fvar,max is the fractional occupation of the statesthat become empty at any given point divided by their max-imum value at a reference point for comparison. The moti-vation of this form of adjustment of the linear-response U isthat it highlights the fact that as n j → 0, χ j → 0 must alsofollow. The approximation to this form is validated if the neteffect of the other consistently occupied states is, on average,

constant. For a reaction coordinate where the maximum andminimum of Nocc is the same number everywhere, the χ andU values are treated on equal footing. However, in cases, suchas MnO2, where the number of occupied, localized-manifoldstates changes, the values of U at the different points are nolonger equivalent. The change in occupied states correspondsstrongly to an evolution from a highly covalent state to a statewith greater ionic character. In Fig. 5, the adjustment on Uand U ↓ produces a value of U around 5.5 eV that is in-variant with respect to O–Mn–O angle variation. These re-sults suggest that the apparent increase in U for linear MnO2

is unphysical, and a potential energy curve calculated at U= 5.5 eV would be most suitable for comparing bent and lin-ear structures. An adjustment to the linear-response U mayprove useful for cases where a large variation in geometricand electronic structure occurs.

ACKNOWLEDGMENTS

The authors gratefully acknowledge helpful conversa-tions with Davide Ceresoli, Matteo Cococcioni, Elise Li, andAdam H. Steeves. We especially want to thank Matteo Co-coccioni for providing us an early release of his DFT+U+Vsubroutines.

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